Parameterized algorithms for multi-label periodic temporal graph realization
In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n×n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between a...
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| Vydáno v: | Theoretical computer science Ročník 1051; s. 115412 |
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Elsevier B.V
09.10.2025
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| Abstract | In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n×n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by D. We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to ℓ times in each period, for some given integer ℓ. For the resulting problem Multi-Label Periodic TGR, we show that it is fixed-parameter tractable for parameter n and for parameter vc+Δ, where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter nu+dmax, where nu is the number of non-universal vertices of the underlying graph and dmax is the largest entry of D. Furthermore, we show that the problem is NP-hard for each ℓ≥5, even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if ℓ=1.
•Introducing the problem of realizing a fastest travel matrix by a periodic temporal graph with ≤ℓ labels per edge.•It is NP-hard even for trees, contrary to the case with a single time label per edge.•It is FPT for parameter n (#vertices) and parameter vc+Δ (vertex cover number + period). |
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| AbstractList | In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n×n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by D. We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to ℓ times in each period, for some given integer ℓ. For the resulting problem Multi-Label Periodic TGR, we show that it is fixed-parameter tractable for parameter n and for parameter vc+Δ, where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter nu+dmax, where nu is the number of non-universal vertices of the underlying graph and dmax is the largest entry of D. Furthermore, we show that the problem is NP-hard for each ℓ≥5, even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if ℓ=1.
•Introducing the problem of realizing a fastest travel matrix by a periodic temporal graph with ≤ℓ labels per edge.•It is NP-hard even for trees, contrary to the case with a single time label per edge.•It is FPT for parameter n (#vertices) and parameter vc+Δ (vertex cover number + period). |
| ArticleNumber | 115412 |
| Author | Morawietz, Nils Wolf, Petra Erlebach, Thomas |
| Author_xml | – sequence: 1 givenname: Thomas orcidid: 0000-0002-4470-5868 surname: Erlebach fullname: Erlebach, Thomas email: thomas.erlebach@durham.ac.uk organization: Department of Computer Science, Durham University, Durham, UK – sequence: 2 givenname: Nils orcidid: 0000-0002-7283-4982 surname: Morawietz fullname: Morawietz, Nils email: nils.morawietz@uni-jena.de organization: Friedrich Schiller University Jena, Institute of Computer Science, Jena, Germany – sequence: 3 givenname: Petra orcidid: 0000-0003-3097-3906 surname: Wolf fullname: Wolf, Petra email: mail@wolfp.net organization: LaBRI, CNRS, Université de Bordeaux, 351, cours de la Libération, Talence, 33405, France |
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| Cites_doi | 10.14778/2732939.2732945 10.1145/3156693 10.1080/17445760.2012.668546 10.1016/j.jcss.2020.08.001 10.1016/0012-365X(91)90068-D 10.1090/qam/184873 10.1007/s00453-018-0478-6 10.1007/978-1-4471-5559-1 10.1007/s00224-017-9757-x 10.1016/0166-218X(92)90041-8 10.1142/S0129054103001728 |
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| Keywords | Almost-clique Temporal graph Dynamic network Fixed-parameter tractability Kernelization |
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| Snippet | In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n×n matrix D of desired fastest travel... |
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| SubjectTerms | Almost-clique Dynamic network Fixed-parameter tractability Kernelization Temporal graph |
| Title | Parameterized algorithms for multi-label periodic temporal graph realization |
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